Welcome back to Physics 211

1
Welcome back to Physics 211

• Todays agenda
• Center of mass
• Equilibrium of extended objects

2
Motion of Real Objects

• So far discussed motion of idealized point-like
  objects
• Saw that neglecting internal forces ok
• only net external forces need to be considered
  for linear motion of center of mass
• what about rotational motion ?

3
Rigid Bodies

• Real extended objects can move in complicated
  ways (stretch, twist etc)
• Here, think of relative positions of each piece
  of body as fixed idealize object as rigid body
• Can still undergo complicated motion (linear
  motion plus rotations)

4
Center of Mass

• Properties
• When a collection of particles making up an
  extended body is acted on by external forces the
  center of mass moves as if all the mass of the
  body were concentrated there.
• weight force can be considered to act vertically
  through center of mass

5
Center of mass for system of (point) objects
6
Points to note

• All real bodies are just collections of
  point-like objects (atoms)
• It is not necessary that CM lie within volume of
  body

7
Two carts, A and B, of different mass (mB 2 mA)
are placed a distance of 90 cm apart. The
location of the center of mass of the two carts is

• 1. 30 cm to the right of cart A.
• 2. 45 cm to the right of cart A.
• 3. 60 cm to the right of cart A.
• 4. None of the above.

8
Two carts, A and B, of different mass (mB 2 mA)
are placed end-to-end on a low-friction track
with a compressed spring between them. After the
spring is released, cart A moves to the left
cart B, to the right. Will the center of mass
of the system

• 1. move to the right,
• 2. move to the left, or
• 3. stay at rest.
• 4. No clue.

9
Use conservation of momentum
m1Dv1m2Dv20 - no external forces
i.e m1v1m2v2const
initially at rest ?const0
i.e m1Dr1m2Dr20 i.e D(m1r1m2r2)0 i.e rCM does
not move!
10
A cart (of mass m) moving to the right at speed v
collides with an identical stationary cart on a
low-friction track. The two carts stick together
after the collision and move to the right with
speed 0.5 v. Is the speed of the center of mass
of the system after the collision the speed of
the center of mass before the collision? 4. Not
sure.

• 1. less than
• 2. equal to, or
• 3. greater than

11
Fnet0
Consider 2 particles and Mm1m2 CM definition
? MrCMm1r1m2r2 MDrCM/Dt m1Dr1/Dtm2Dr2/Dt
m1v1m2v2 RHS is total momentum Thus velocity of
center of mass is constant in absence of external
forces !
12
Conclusions

• If there is no net force on a system, the center
  of mass of the system

• will stay at rest if it is initially at rest, or
• will continue to move with the same velocity if
  it is initially moving.

13
What about Fext not zero ?
M rCM m1r1m2r2 ? MD rCM / Dt m1Dr1/ Dt
m2Dr2/ Dt ? MvCM m1v1m2v2 Therefore
M DvCM / Dt m1 Dv1 / Dt m2 Dv2 / Dt M aCM
F1F2 Fext since internal forces cancel.
14
Motion of center of mass of a system
The center of mass of a system of point objects
moves in the same way as a single object with the
same total mass would move under the influence of
the same net (external) force.
15
Throwing a wrench

• Demo wrench 1 point has simple motion
  projectile - center of mass
• Total external force Fext
• aCM Fext /M
• Linear motion of system looks like all mass is
  concentrated at CM

16
Motion of center of mass of a system

• Since extended objects are just systems of large
  point objects (atoms) this also holds for them
• How does one find the center of mass of an
  extended object?

17
Center of mass of extended object
For simple geometric objects (e.g., circle,
rectangle, etc.) the center of mass is at the
geometric center.
18
Equilibrium of extended object

• Clearly net force must be zero
• Also, if want object to behave as point at center
  of mass ? ALL forces acting on object must pass
  through CM